LECTURE 0: ESSENTIAL PRIOR KNOWLEDGE FOR BIOEN 326:
Note that this is not today’s lecture notes; Lecture 1 and Homework 1 are posted on the web page, on today’s
date on the schedule page. See the course homepage: http://courses.washington.edu/bioen326/index.htm. You
should have a link to this through myUW.
Prerequiste Knowledge:
I assume you have taken classes that are prerequisites for the prerequisites. This includes knowledge from:
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CHEM 142-162 (chemical reactions, thermodynamics, molecular bonds)
PHYS 121 (friction, force, moments, free body diagrams, newton’s laws)
MATH 126, (introduction to multivariable differential calculus)
MATH 307, solutions to linear differential equations
MATH 308, especially matrix algebra
AMATH 301: numerical solutions to ODEs and matrix algebra.
BIOEN 315: protein structure and energy, thermodynamics, kinetics.
Ordinary differential equations (ODEs)
For this class you should be able to solve linear ODEs. When you have a simultaneous system of more than
one ODE, you need to know how to convert to one or solve two simultaneously. You should know how to
solve these using Laplace transforms. Recall the following:
For first order ODEs, you need one ODE and one initial condition for each dependent variable. For second
order ODEs, you need the initial condition for both the function and its derivative. (We are not addressing
boundary value problems here, although we will touch on these briefly when we discuss buckling of columns).
The Complementary solution is when the initial Condition holds, but there is no pushing function. The
Particular solution is when there is a Pushing function, but the initial conditions are zero. Since the equations
are linear, the overall solution is the sum of the response to the Initial conditions and the pushing function, or
the complementary plus particular solution.
To use Laplace transforms
∞
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you can define 𝑌(𝑠) = 𝐿(𝑦(𝑡)) = ∫0 𝑦(𝑡)𝑒 −𝑠𝑡 𝑑𝑡. We often abbreviate 𝑌 = 𝐿(𝑦).
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Then, 𝐿 ( ) = 𝑠𝑌(𝑠) − 𝑦(0), where 𝑦(0) is the initial condition.
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Some common transform pairs are:
o 𝐿(𝛿(𝑡)) = 1 (impulse function)
o 𝐿(∅(𝑡)) = 1/𝑠 (step function; 0 below t = 0, and 1 above t = 0)
o 𝐿(𝑡∅(𝑡)) = 1/𝑠 2

𝑑𝑦
𝑑𝑡
1
o
𝐿(𝑒 −𝑎𝑡 ∅(𝑡)) = 𝑎+𝑠
o
𝐿(sin(𝑎𝑡)) = 𝑎2 +𝑠2
𝑎
o 𝐿(𝑏1 𝑦1 + 𝑏2 𝑦2 ) = 𝑏1 𝐿(𝑦1 ) + 𝑏2 𝐿(𝑦2 ) (linear property of Laplace transforms)
Method to apply Laplace transforms:
o transform both sides of the ODE. You may want to do this for the Particular solution only, so
you can set all initial conditions to zero.
o manipulate to get Y = G(s), where G is a polynomial in s.
o Use
partial
fraction
decomposition
(http://en.wikibooks.org/wiki/Complex_Analysis/Residue_Theory/Partial_Fractions )or method
of
residues
(also
called
Heaviside
cover
up
method)
(http://www.math.vt.edu/people/dlr/m2k_opm_lappf2.pdf
or
o
o
http://en.wikipedia.org/wiki/Heaviside_cover-up_method) to manipulate G(s) to be a sum of
terms that are known transform pairs (eg on the right side of the transform pair list above).
Take the inverse transform
If you solved for the particular solution, use a different method to solve for the complementary
solution and add it here.
Partial Derivatives
Recall that a partial derivative is indicated by the del operator, as in:
𝜕𝑢
𝜕𝑥
and refers to the derivative of the
variable in the numerator with respect to that in the denominator while holding all other variables constant.
For example, if 𝑢 = 𝑦 2 + 𝑧 ∗ 𝑠𝑖𝑛(𝑥), then
𝜕𝑢
𝜕𝑦
= 2𝑦 ,
𝜕𝑢
𝜕𝑥
= 𝑧𝑐𝑜𝑠(𝑥),
𝜕𝑢
𝜕𝑧
= 𝑠𝑖𝑛(𝑥). You will use these a lot in 325,
but we will touch on them here.
Mechanics (most of phys 121 or equivalent).
Definitions
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a force, 𝐹⃑ is a vector applied at a point.
⃑⃑⃑ = 𝑟⃑ × 𝐹⃑ , where 𝑀
⃑⃑⃑
a moment results when you consider the effect of this force on a second point O: 𝑀
is the moment at point O, and 𝑟⃑ is the vector from O to where the force is applied. Recall that the cross
product is perpendicular to the plane of the two vectors involved, so comes out of the paper in this
example. Specifically, 𝑟⃑ × 𝐹⃑ = 𝑟𝐹𝑠𝑖𝑛𝜃 ∙ 𝑛⃑⃑, where 𝑛⃑⃑ is the unit vector with direction determined by the
right hand rule, and r is the distance between O and where force is applied, while F is the magnitude of
the force. When 𝑟⃑ and 𝐹⃑ are in-line, then 𝑠𝑖𝑛𝜃 = 0, so there is no moment. The moment is maximum
when the two are perpendicular. Finally, the moment increases with the distance r and force F.
Free body diagrams. Recall that an object at equilibrium is either at rest or has a constant linear and angular
velocity. This occurs when all forces and moments on the object are balanced, but in phys 121, you focused on
problem where there were no moments so we just review this. To use a free body diagram to calculate forces:
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Draw the object, indicating distances and angles of interest, whether known or unknown.
Indicate external forces acting on the object. These may be known or unknown:
Write down the knowns and unknowns to clarify which are which.
Write the equilibrium equations. At equilibrium, for every point on the object, the forces in each
direction must sum to zero. In 2D, ∑ 𝐹𝑥 = 0 , ∑ 𝐹𝑦 = 0 .
Solve for the unknowns. You should have one independent equation for every unknown, or the
problem is not solvable.
Other:
Linear algebra: You should be able to multiply arrays
Chemical thermodynamics: you should know the Boltzman distribution, which states that the probability
𝑒
of a molecule (or similar system) being in state 1 is: 𝑝1 ∝ exp(− 𝑘 1𝑇), where 𝑒1 is the energy of the element
𝐵
𝐸
in that state. Alternatively, you may be familiar with the formula: 𝑃1 ∝ exp(− 𝑅𝑇1 ), where 𝑃1 is the fraction of
molecules in state 1, and 𝐸1 is the energy of a mole of molecules in that state. However, we will go over this
again in class, so this prior knowledge is not essential.
Mathematical Programming: you should be able to use MATLAB or a similar mathematical programming
software to solve linear algebra and ODE problems. You should be able to:
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write and run an m-file script
write m-file functions to be called
define variables of interest
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use matrix algebra to multiply arrays
use ODE solvers such as ode23s to solve ODE problems.
make figures to plot results
Test Yourself! Can you solve these Review Problems?
(some will be included in homeworks when we need the skill)
1.
Ordinary differential equations (ODE): given the equations
𝑑𝑦
𝑑𝑡
= −𝑎𝑦 + 𝑏, and 𝑦(0) = 𝑐,
a. what is the complementary solution to this problem?
b. what is the particular solution?
c. what is the overall solution, y(t)?
d. verify your result by plugging the solution back into the original equations.
2. partial derivatives:
𝑥
𝑦
a. given the function, 𝑢 = , what are
𝜕𝑢
𝜕𝑥
and
𝜕𝑢
?
𝜕𝑦
3. mechanics:
a. draw the free body diagram for the following problem (from Tipler problem 4-52c): For the
system in the following figure to be in equilibrium, find the unknown tensions T1, T2, and T3
and mass m. (The weight is 5 kg in case you can’t read it; the bar is horizontal.)
4. linear algebra
𝑒
𝑒
𝑎 𝑏
𝑎 𝑏
a. what is [
] ∗ [𝑓 ]? what is [𝑓 ] ∗ [
]?
𝑐 𝑑
𝑐 𝑑
5. chemical thermodynamics (optional; you will learn this in class, this is just to jog your memory)
a. What is the relationship of the Boltzman constant, 𝑘𝐵 , to the gas constant, 𝑅?
b. How are these used to calculate the probability of a molecules being in a state?
c. If a molecule can have two states, one with energy E1 and the other with E2, what is the
probability of being in each state?
6. mathematical programming: use a mathematical program such as MATLAB to find the numerical
solution to the following, and verify that this is the same as your analytic solution obtained above:
a. problem 1d, with a = 2, b = 5 and c = 0 (remember how to use ODE solvers?)
b. problem 3a, with a = b = 1, c = 2, d = 0.5, e = 1, f = 1.